Heat Transfer Theoretical Analysis Experimental Investigations Systems Part 13 pot

Heat Transfer Phenomena in Laminar Wavy Falling Films: Thermal Entry Length, Thermal-Capillary Metastable Structures, Thermal-Capillary Breakdown 471 Therefore, in Fig.. 13 the experime

Approximating this equation by a finite differential equation the typical temperature drop

over lx≡0.9λW can be determined:

Where ′′q is the heat flux at the wall, UW res is the average liquid velocity in the residual layer,

ρ is the liquid density and cp is the specific heat The remaining ten percent of lx correspond

to the length of the wave front The material properties are taken at inflow temperatures

From the thermographic pictures as presented in Fig 7.c.1, the temperature difference in x

and y direction can be evaluated Thereby a proportionality can be determined

⋅

As for current data the constant of proportionality is in the range between 1 < k < 5 For

further considerations a value of k = 3 was assumed

According to expression (20) the critical heat flux depends on the liquid properties, the

frequency of large waves and the typical transverse size of regular structures If the

following dimensionless numbers are used:

Heat Transfer Phenomena in Laminar Wavy Falling Films:

Thermal Entry Length, Thermal-Capillary Metastable Structures, Thermal-Capillary Breakdown 471

Therefore, in Fig 13 the experimental data for the dimensionless critical heat flux

obtained on excited falling films (with the help of the loud speaker) were used, allowing to

keep the major frequency fW at a constant value As can be seen Eq (22) depends on the

relation between the mean velocity of the residual layer and the mean velocity of large

waves In the literature many different analytical and empirical equations can be found for

these velocities as functions of the Reynolds number For example in (Brauner & Maron,

1983) a physical model for the falling film is presented In this case the ratio is constant:

res W

Equation (24) is in the same order for the dimensionless critical heat flux as the experimental

data, but the trend of the latter has a different inclination, see Fig 13

In (Al-Sibai, 2004) the same silicone oils were used as in the current experiments Therefore a

better comparability could be given for dependencies from (Al-Sibai, 2004) as for other

correlations from literature Since the thickness of the residual layer is relatively small the

Nusselt formula for laminar flow can be used:

ν

2 res resgδ

Re

0 6 2

0 36

Re

.

Fig 13 The dependence of dimensionless parameter Maq/(PrKΛ) versus Reynolds-number

Points experimental data (Lel et al., 2007a)

3 3 res

3 3 W

1 + 0.219ReMa

= 5.6 × 10

The comparison of dependence (29) with experimental data gives a good agreement in the

order of magnitude but a difference in the inclination, see Fig 13 For a Reynolds number

range Re < 3 the dimensionless parameter (Maq/PrKΛ) according to Eq (29) even decreases,

but the experimental data show another tendency

This disagreement between experimental data and theory can be ascribed to the uncertainty

of the proportionality factor in Eq (16) as describes above

As can be seen from (29):

Ma = f Re,Pr,K (30) This approximation found from experimental data analysis is:

-4 1.44

Ma = 5.59× 10 Re PrK (31)

In order to verify and consolidate this theory the range of Reynolds number should be

increased An elongating of heat section will allows the observation of further development

of regular structures

For the experiments without activated loud speaker the wavelength has to be determined by

measuring the oscillations of the film surface and using the major frequency for the

parameter KΛ

A comparison of experimental data with other dependencies from the literature is shown in

the next part

4.2 Comparing of experimental data with other approachs

In this part different approaches for the determination of the critical dimensionless heat flux

are presented and compared with experimental data

Experimental data for laminar-wavy and turbulent films were described in (Gimbutis, 1988)

by the following empirical dependencies:

Heat Transfer Phenomena in Laminar Wavy Falling Films:

Thermal Entry Length, Thermal-Capillary Metastable Structures, Thermal-Capillary Breakdown 473

q

Re

For 100 < Re < 200 in (Gimbutis, 1988) the scattering of data was up to 50 %, for Re < 100 no

experimental data have been recorded It can be seen in Fig 14 that this dependence

suggests lower values than the current experimental data The difference can be explained

by the fact that in (Gimbutis, 1988) the experimental data were obtained only for a water

film flow with a relatively long heated section In this case evaporation effects and thus a

shift in the thermophysical properties could have appeared

In (Kabov, 2000) the empirical dependence of the critical Marangoni number on the

Reynolds number for a shorter heated section (6.5 mm length along the flow) for laminar

waveless falling films was obtained:

0.98 q

In this case the length of the heated section is in the same order of magnitude as the thermal

entry length (Kabov, 2000) Therefore this curve indicates higher values than our

100

1000

6420

Fig 14 The dependence of dimensionless parameter Maq versus the Reynolds-number

Points experimental data (Lel et al., 2007a)

It was shown in (Ito et al., 1995) that for the 2D case the modified critical Marangoni number

4/3 q

It can be seen, that only dependence (35) is in the same order of magnitude as our

experimental data

Other dimensionless parameters for generalisation of experimental data were used in (Bohn

& Davis, 1993) and (Zaitsev et al., 2004) In (Bohn & Davis, 1993) the data for dimensionless

breakdown heat flux is approximated in the form:

Ma

= 4.78×10 Re

Fig 15 shows that (37) again leads to lower values than our experimental data Here, as in

case of Eq (32), evaporation effects could have appeared, because this dependence was

obtained for water and for a 30 % glycerol-water solution at a 2.5 m long test section for

Re > 959

Re

DMS-T11 DMS-T20

Fig 15 The dependence of dimensionless parameter Maq/Pr versus the Reynolds-number

Points experimental data (Lel et al., 2007a)

A generalisation for water and an aqueous solution of alcohol is presented in (Zaitsev et al.,

This correlation leads to results which exceed current data by more than one order of

magnitude This can be partially explained by the fact that dependence (38) was obtained for

Heat Transfer Phenomena in Laminar Wavy Falling Films:

Thermal Entry Length, Thermal-Capillary Metastable Structures, Thermal-Capillary Breakdown 475

stable dry spots, whereas the new data was recorded for the formation of local instable dry

spots

5 Thermal entry length

In this part experimental data for the thermal entry length with the correlation from the

literature are compared and a new correlation, included dimensionless parameters

incorporated several physical effects, is presented

A comparison of experimental data for the thermal entry length with correlations for

laminar flow by (Mitrovic, 1988) and (Nakoryakov & Grigorijewa, 1980), is shown in Fig 16

Whereas for very low Reynolds numbers (Re0 < 3) and heat fluxes the experimental data

correlate satisfactorily with these dependencies, at larger Reynolds numbers and heat fluxes

the experimental values lie under the ones obtained through correlations

Fig 16 Experimental data compared with solutions for smooth laminar falling films [1] –

experimental data by (Lel et al., 2007b); [2] – experimental data by (Lel et al., 2009)

Therefore, the experimental data were used in order to found a empirical dependency which

describes the dimensionless entry length and attempts to incorporate several effects: i) the

effect of nonlinear changing material properties due to temperature changes and the effects

of ii) surface tension and iii) waves:

In Fig 17 the comparison of experimental data with a correlation for 100<Pr0<180 and 2<Re0<40 is presented

Here it is significant, that the differences of Eq (1) and Eq (39) are the additional terms involving Pr0/PrW and the Kapitza number Ka0 = (σ3ρ/gη4)

(Brauer, 1956) found that the Kapitza number has an influence on the development of the

The effect of the decrease of the thermal entry length flux because of Marangoni convection, described in (Kabov et al., 1996), is subject to debate We assume that in this case the influence of the waves on the thermal entry length play a dominant role This question stays unsettled and should be investigated in future

By comparing the temperature and film thickness fields, the assumption of the capillary nature (Marangoni effect) of regular structures within the residual layer has been

thermo-Heat Transfer Phenomena in Laminar Wavy Falling Films:

Thermal Entry Length, Thermal-Capillary Metastable Structures, Thermal-Capillary Breakdown 477 confirmed An increase in local surface temperature leads to a decrease in local film thickness The evolution of the regular structure’s “head” between two large parabolic shaped waves over time was presented

The decrease of the mean film thickness could be explained by a reduction of the viscosity and a cross flow into the faster moving large waves Both effects cause a higher film velocity

The results obtained are important for the investigation of the dependency between wave characteristics and local heat transfer, the conditions of “dry spot” appearance and the development of crisis modes in laminar-wavy falling films

A model of thermal-capillary breakdown of a liquid film and dry spot formation is suggested on the basis of a simplified force balance considering thermal-capillary forces in the residual layer It is shown that the critical heat flux depends on half the distance between two hot structures, because the fluid within the residual layer is transferred from hot structures to the cold areas in between them It also depends on the main frequency of large waves, the Prandtl number, the heat conductivity, the liquid density and the change in surface tension in dependence on temperature The model is also presented in a dimensionless form

The investigations of the thermal entry length of laminar wavy falling films by means of infrared thermography are shown Good qualitative agreement with previous works on laminar and laminar wavy film flow was found at low Reynolds numbers However, with increasing Reynolds numbers and heat fluxes, these correlations describe the thermal entry length inadequately The correlation established for laminar flow was extended in order to include the effect of temperature-dependent non-linear material properties as well as for the effects of surface tension and waves

7 Acknowledgements

This work was financially supported by the “Deutsche Forschungsgemeinschaft” (DFG

KN 764/3-1) The authors thank the student coworkers and colleagues A Kellermann, Dr

H Stadler, Dr G Dietze, M Baltzer, Dr F Al-Sibai, M Allekotte for the help in the preparation of this chapter

8 References

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Symp on Two-Phase Flow Modeling and Experimentation, Vol.1, pp 203–210, Rome, Italy, 1996

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19

Heat Transfer to Fluids at Supercritical Pressures

Igor Pioro and Sarah Mokry

University of Ontario Institute of Technology

Canada

1 Introduction

Prior to a general discussion on parametric trends in heat transfer to supercritical fluids, it is important to define special terms and expressions used at these conditions Therefore, general definitions of selected terms and expressions, related to heat transfer to fluids at critical and supercritical pressures, are listed below For better understanding of these terms and expressions a graph is shown in Fig 1 General definitions of selected terms and expressions related to critical and supercritical regions are listed in the Chapter

“Thermophysical Properties at Critical and Supercritical Conditions”

Axial Location, m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

o C

300 350 400 450

600 550 500

Bulk Fluid Enthalpy, kJ/kg

Heat transfer coefficient

pin=24.0 MPa G=503 kg/m 2 s Q=54 kW

qave= 432 kW/m 2

C 381.1

Fig 1 Temperature and heat transfer coefficient profiles along heated length of vertical

circular tube (Kirillov et al., 2003): Water, D=10 mm and L h=4 m

General definitions of selected terms and expressions related to heat transfer at critical and supercritical pressures

Deteriorated Heat Transfer (DHT) is characterized with lower values of the wall heat

transfer coefficient compared to those at the normal heat transfer; and hence has higher values of wall temperature within some part of a test section or within the entire test section

Improved Heat Transfer (IHT) is characterized with higher values of the wall heat transfer

coefficient compared to those at the normal heat transfer; and hence lower values of wall temperature within some part of a test section or within the entire test section In our opinion, the improved heat-transfer regime or mode includes peaks or “humps” in the heat

transfer coefficient near the critical or pseudocritical points

Normal Heat Transfer (NHT) can be characterized in general with wall heat transfer

coefficients similar to those of subcritical convective heat transfer far from the critical or pseudocritical regions, when are calculated according to the conventional single-phase

Dittus-Boelter-type correlations: Nu = 0.0023 Re0.8Pr0.4

Pseudo-boiling is a physical phenomenon similar to subcritical pressure nucleate boiling,

which may appear at supercritical pressures Due to heating of supercritical fluid with a bulk-fluid temperature below the pseudocritical temperature (high-density fluid, i.e.,

“liquid”), some layers near a heating surface may attain temperatures above the pseudocritical temperature (low-density fluid, i.e., “gas”) (for specifics of thermophysical properties, see Chapter “Thermophysical Properties at Critical and Supercritical Conditions”) This low-density “gas” leaves the heating surface in the form of variable density (bubble) volumes During the pseudo-boiling, the wall heat transfer coefficient usually increases (improved heat-transfer regime)

Pseudo-film boiling is a physical phenomenon similar to subcritical-pressure film boiling,

which may appear at supercritical pressures At pseudo-film boiling, a low-density fluid (a fluid at temperatures above the pseudocritical temperature, i.e., “gas”) prevents a high-density fluid (a fluid at temperatures below the pseudocritical temperature, i.e., “liquid”) from contacting (“rewetting”) a heated surface (for specifics of thermophysical properties, see Chapter “Thermophysical Properties at Critical and Supercritical Conditions”) Pseudo-film boiling leads to the deteriorated heat-transfer regime

Water is the most widely used coolant or working fluid at supercritical pressures The largest application of supercritical water is in supercritical “steam” generators and turbines, which are widely used in the power industry worldwide (Pioro and Duffey, 2007) Currently, upper limits of pressures and temperatures used in the power industry are about 30 – 35 MPa and 600 – 625ºC, respectively New direction in supercritical-water application in the power industry is a development of SuperCritical Water-cooled nuclear Reactor (SCWR) concepts, as part of the Generation-IV International Forum (GIF) initiative However, other areas of using supercritical water exist (Pioro and Duffey, 2007)

Supercritical carbon dioxide was mostly used as a modelling fluid instead of water due to significantly lower critical parameters (for details, see Chapter “Thermophysical Properties

at Critical and Supercritical Conditions”) However, currently new areas of using supercritical carbon dioxide as a coolant or working fluid have been emerged (Pioro and Duffey, 2007)

Heat Transfer to Fluids at Supercritical Pressures 483 The third supercritical fluid used in some special technical applications is helium (Pioro and Duffey, 2007) Supercritical helium is used in cooling coils of superconducting electromagnets, superconducting electronics and power-transmission equipment

Also, refrigerant R-134a is being considered as a perspective modelling fluid due to its lower critical parameters compared to those of water (Pioro and Duffey, 2007)

Experiments at supercritical pressures are very expensive and require sophisticated equipment and measuring techniques Therefore, some of these studies (for example, heat transfer in bundles) are proprietary and hence, were not published in the open literature The majority of studies (Pioro and Duffey, 2007) deal with heat transfer and hydraulic resistance of working fluids, mainly water, carbon dioxide and helium, in circular bare tubes In addition to these fluids, forced- and free-convection heat-transfer experiments were conducted at supercritical pressures, using liquefied gases such as air, argon, hydrogen; nitrogen, nitrogen tetra-oxide, oxygen and sulphur hexafluoride; alcohols such as ethanol and methanol; hydrocarbons such as n-heptane, n-hexane, di-iso-propyl-cyclo-hexane, n-octane, iso-butane, iso-pentane and n-pentane; aromatic hydrocarbons such as benzene and toluene, and poly-methyl-phenyl-siloxane; hydrocarbon coolants such as kerosene, TS-1 and RG-1, jet propulsion fuels RT and T-6; and refrigerants

A limited number of studies were devoted to heat transfer and pressure drop in annuli, rectangular-shaped channels and bundles

Accounting that supercritical water and carbon dioxide are the most widely used fluids and that the majority of experiments were performed in circular tubes, specifics of heat transfer and pressure drop, including generalized correlations, will be discussed in this chapter based on these conditions1

Specifics of thermophysical properties at critical and supercritical pressures for these fluids are discussed in the Chapter “Thermophysical Properties at Critical and Supercritical Conditions” and Pioro and Duffey (2007)

2 Convective heat transfer to fluids at supercritical pressures: Specifics of supercritical heat transfer

flowing inside circular tubes at supercritical pressures are listed in Pioro and Duffey (2007)

In general, three major heat-transfer regimes (for their definitions, see above) can be noticed

at critical and supercritical pressures (for details, see Figs 1 and 2):

1 Normal heat transfer;

2 Improved heat transfer; and

3 Deteriorated heat transfer

Also, two special phenomena (for their definitions, see above) may appear along a heated surface:

These heat-transfer regimes and special phenomena appear to be due to significant

variations of thermophysical properties near the critical and pseudocritical points (see Fig

3) and due to operating conditions

Axial Location, m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

o C

350 375 400

2 K

20 30 40 50 60 70 80

Bulk Fluid Enthalpy, kJ/kg

Heated length

Bulk fluid temperatur

e Inside wall temperature

Heat transf

er coefficient

pin=24.0 MPa, G=1494 kg/m 2

s, Q=61 kW, qave=489 kW/m 2

ure

Heat transfer coefficient

pin=23.9 MPa, G=997 kg/m 2

s Q=74 kW, qave= 584 kW/m 2

C 380.8

pc =

HpcDittus - Boelter correlation

DHT Normal HT

Axial Location, m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

550 500

Bulk Fluid Enthalpy, kJ/kg

Heat transfer coefficient

pin=24.0 MPa G=1000 kg/m 2 s Q=103 kW qave= 826 kW/m 2

C 381.3

pc =

H pc Dittus - Boelter correlation

DHT

(b) (c) Fig 2 Temperature and heat transfer coefficient profiles along heated length of vertical

circular tube (Kirillov et al 2003): Water, D=10 mm and L h=4 m

Therefore, the following cases can be distinguished at critical and supercritical pressures (for

details, see Figs 1 and 2):

a Wall and bulk-fluid temperatures are below a pseudocritical temperature within a part

or the entire heated channel;

b Wall temperature is above and bulk-fluid temperature is below a pseudocritical

temperature within a part or the entire heated channel;

Heat Transfer to Fluids at Supercritical Pressures 485

c Wall temperature and bulk fluid temperature is above a pseudocritical temperature within a part or the entire heated channel;

d High heat fluxes;

e Entrance region;

f Upward and downward flows;

g Horizontal flows;

h Effect of gravitational forces at lower mass fluxes; etc

All these cases can affect the supercritical heat transfer

0 10 30 40 60 80 100 120

25 30 35 40 45 50 55 60 65 70 75 80

Bulk Fluid Enthapy, kJ/kg

Heated length

C 381.3

Fig 3 Temperature and thermophysical properties profiles along heated length of vertical

circular tube (operating conditions in this figure correspond to those in Fig 2c): Water, D=10

mm and L h=4 m; thermophysical properties based on bulk-fluid temperature

3 Parametric trends

3.1 General heat transfer

As it was mentioned above, some researchers suggested that variations in thermophysical properties near critical and pseudocritical points resulted in the maximum value of Heat Transfer Coefficient (HTC) Thus, Yamagata et al (1972) found that for water flowing in vertical and horizontal tubes, the HTC increases significantly within the pseudocritical region (Fig 4) The magnitude of the peak in the HTC decreases with increasing heat flux and pressure The maximum HTC values correspond to a bulk-fluid enthalpy, which is slightly less than the pseudocritical bulk-fluid enthalpy

Results of Styrikovich et al (1967) are shown in Fig 5 Improved and deteriorated transfer regimes as well as a peak (“hump”) in HTC near the pseudocritical point are clearly shown in this figure The deteriorated heat-transfer regime appears within the middle part

heat-of the test section at a heat flux heat-of about 640 kW/m2, and it may exist together with the improved heat-transfer regime at certain conditions (also see Fig 1) With the further heat-flux increase, the improved heat-transfer regime is eventually replaced with that of deteriorated heat transfer

o C

200 400

fer oe ci ts

r coficie

various pressures (Yamagata et al., 1972): Water – (a) p=22.6 MPa; (b) p=24.5 MPa; and (c)

p=29.4 MPa

Vikhrev et al (1971, 1967) found that at a mass flux of 495 kg/m2s, two types of deteriorated

heat transfer existed (Fig 6): The first type appeared within the entrance region of the tube L

/ D < 40 – 60; and the second type appeared at any section of the tube, but only within a

certain enthalpy range In general, the deteriorated heat transfer occurred at high heat

fluxes

The first type of deteriorated heat transfer observed was due to the flow structure within the

entrance region of the tube However, this type of deteriorated heat transfer occurred

mainly at low mass fluxes and at high heat fluxes (Fig 6a,b) and eventually disappeared at

high mass fluxes (Fig 6c,d)

Heat Transfer to Fluids at Supercritical Pressures 487

Bulk Fluid Enthalpy, kJ/kg

o C

270 330 360

348 523

640 756 872

Dete riorated

Heat Transfer

heat transfer appeared when q / G > 0.4 kJ/kg (where q is in kW/m2 and G is in kg/m2s)

This value is close to that suggested by Styrikovich et al (1967) (q / G > 0.49 kJ/kg)

However, the above-mentioned definitions of two types of deteriorated heat transfer are not enough for their clear identification

3.2 Pseudo-boiling and pseudo-film boiling phenomena

Ackerman (1970) investigated heat transfer to water at supercritical pressures flowing in smooth vertical tubes with and without internal ribs within a wide range of pressures, mass fluxes, heat fluxes and diameters He found that pseudo-boiling phenomenon could occur

at supercritical pressures The pseudo-boiling phenomenon is thought to be due to large differences in fluid density below the pseudocritical point (high-density fluid, i.e., “liquid”) and beyond (low-density fluid, i.e., “gas”) This heat-transfer phenomenon was affected with pressure, bulk-fluid temperature, mass flux, heat flux and tube diameter

The process of pseudo-film boiling (i.e., low-density fluid prevents high-density fluid from

“rewetting” a heated surface) is similar to film boiling, which occurs at subcritical pressures Pseudo-film boiling leads to the deteriorated heat transfer However, the pseudo-film boiling phenomenon may not be the only reason for deteriorated heat transfer Ackerman noted that unpredictable heat-transfer performance was sometimes observed when the pseudocritical temperature of the fluid was between the bulk-fluid temperature and the heated surface temperature

Kafengaus (1986, 1975), while analyzing data of various fluids (water, ethyl and methyl alcohols, heptane, etc.), suggested a mechanism for “pseudo-boiling” that accompanies heat transfer to liquids flowing in small-diameter tubes at supercritical pressures The onset of pseudo-boiling was assumed to be associated with the breakdown of a low-density wall layer that was present at an above-pseudocritical temperature, and with the entrainment of

individual volumes of the low-density fluid into the cooler (below pseudocritical

temperature) core of the high-density flow, where these low-density volumes collapse with

the generation of pressure pulses At certain conditions, the frequency of these pulses can

coincide with the frequency of the fluid column in the tube, resulting in resonance and in a

rapid rise in the amplitude of pressure fluctuations This theory was supported with

ll temperatures

D=20.4 mm, L=6 m Heat Transfer Coefficient

(c) (d) Fig 6 Temperature profiles (a) and (c) and HTC values (b) and (d) along heated length of a

vertical tube (Vikhrev et al., 1967): HTC values were calculated by the authors of the current

chapter using the data from the corresponding figure; several test series were combined in

each curve in figures (c) and (d)